Stochastic Equilibrium Research Articles

Investing in power grid infrastructure as a flexibility option: A DSGE assessment for Germany

This paper provides an approach to incorporate planned investments in power grid infrastructure in Germany, which are expected to offer the necessary flexibility to integrate large shares of variable renewable energy sources into the power system, into a dynamic stochastic equilibrium model. Since the investments' economic impact remains unclear, our research sheds light on two questions: Do power grid infrastructure investments in Germany have the potential to positively impact economic performance, particularly GDP and employment? Is power grid infrastructure investment an efficient way to provide flexibility to the electricity system? We find the potential of negative effects of power grid infrastructure investments on economic outcomes, which can, however, be mitigated by an adequate design of the investments and its framework conditions.

Dynamical Behavior of Stochastic Markov Switching Hepatitis B Epidemic Model with Saturated Incidence Rate

The article researches a stochastic hepatitis B epidemic model with saturated incidence rate, which is perturbed by both white noise and colored noise. Firstly, we obtain a significant criterion R 0 S which relies on environmental noises. By means of Lyapunov function approach, we show that there is a stationary distribution if R 0 S > 1 . Its condition implies that when white noise is small, in the stochastic model, there exists a stochastic positive equilibrium state without changing the basic properties of its corresponding deterministic model. Secondly, we derive sufficient criteria for extinction of the disease. Finally, we propose a definition of the solution to an impulsive stochastic functional differential equation with Markovian switching (ISFDM).

Propagation of stochastic travelling waves of cooperative systems with noise

<p style='text-indent:20px;'>We consider the cooperative system driven by a multiplicative It\^o type white noise. The existence and their approximations of the travelling wave solutions are proven. With a moderately strong noise, the travelling wave solutions are constricted by choosing a suitable marker of wavefront. Moreover, the stochastic Feynman-Kac formula, sup-solution, sub-solution and equilibrium points of the dynamical system corresponding to the stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.</p>

Risk aversion in multilevel electricity market models with different congestion pricing regimes

Due to ongoing efforts for decarbonization, electricity markets worldwide are undergoing fundamental transitions, which result in increased uncertainty for all market participants. Against this background, we investigate the impact of risk aversion on investment and market operation in markets with different congestion pricing regimes and multi-level decision making. We develop a stochastic multi-level equilibrium model with risk-averse agents, which includes investment in transmission and generation capacity, market operation, and redispatch. The model can incorporate perfect, as well as imperfect locational price signals and different upper-level expectations about lower-level risk aversion. We apply our model to a stylized two-node example and compare the effects of risk aversion in a system with zonal and nodal pricing, respectively. Our results show that the effect of risk aversion is more pronounced in a market with nodal pricing, compared to a market with imperfect locational price signals. Furthermore, transmission planners that are ignorant about risk aversion of generation companies can induce substantial additional costs, especially in a nodal pricing market.

The impact of the minimum housing scale constraint on life-cycle risky asset and housing investment

While the standard loan-to-value (LTV) and loan-to-income (LTI) mortgage borrowing constraints are widely studied in the theoretical housing demand literature, the way they interact with the minimum housing scale (MHS) constraint to determine life-cycle choices has received far less attention. We adopt a stochastic partial equilibrium life-cycle model within which consumers choose a portfolio of three assets, subject to standard LTV and LTI constraints, as well as an MHS. We highlight features of possible solutions involving different binding constraints each period. Moreover, we provide closed-form solutions and simulated life time paths for different realisations of uncertainty with Stone-Geary preferences. Our simulation results show that the MHS constraint generates an increasing profile of risky asset share profile in the presence of mortgage constraints, and imply that the MHS may worsen poor people’s welfare by crowding them out of home ownership for a longer period of their life-time and exacerbate the wealth inequality both in risky asset and housing wealth in the long run.

A two-stage stochastic variational inequality model for storage and dynamic distribution of medical supplies in epidemic management

The storage and distribution of medical supplies are important parts of epidemic prevention and control. This paper first proposes a new nonsmooth two-stage stochastic equilibrium model of medical supplies in epidemic management. The first stage addresses the storage in the pre-disaster phase, and the second stage focuses on the dynamic distribution by enrolling competitions among multiple hospitals over a period of time in the post-disaster phase. The uncertainties are the numbers of infected people treated in multiple hospitals during the period of time, which are time-varying around a nominal distribution predicted by historical experience. The two-stage stochastic equilibrium model is further approximated and transformed to a monotone two-stage stochastic variational inequality (SVI) model that is computationally tractable, with the aid of a smooth approximation technique. We employ the progressive hedging method (PHM) to solve a case study in the city of Wuhan in China suffered from the COVID-19 pandemic. Numerical results are presented to demonstrate the effectiveness of the proposed model in planning the storage and dynamic distribution of medical supplies in epidemic management.

Optimal time-differentiated pricing for a competitive mixed traditional and crowdsourced event parking market

An event-based parking pricing problem, the Crowdsourced Event Parking Market Pricing Problem, is proposed wherein parking lot owners and others who are willing to rent out privately owned spaces compete to attract drivers who are looking for available parking spaces. Each parking location owner’s problem is modeled as a bi-level program, where the upper-level parking garages, individuals and consolidators (the players) compete for customers, setting their prices to maximize revenue given the response of the lower-level followers. The lower-level followers choose their parking locations based on the utilities they derive from the spaces, which is a function of the proximity of the spaces to their destinations, parking fees and crowdedness. Prices are set as a function of the time of reservation. Reservation-time based pricing enables price differentiation for late comers either in the form of lower prices to attract last-minute customers when excess spaces are anticipated or higher-pricing if few spaces are expected to remain empty. A multi-period, stochastic Equilibrium Problem with Equilibrium Constraints (EPEC) formulation of the Crowdsourced Event Parking Market Pricing Problem is presented, and a diagonalization method with embedded gradient ascent approach for solution of individual player Mathematical Programs with Equilibrium Constraints (MPECs) is proposed for its solution. Both supply- and demand-side uncertainties are explicitly modeled. Solutions provide competitive parking prices set by reservation time for each parking facility, whether the facility involves a large parking garage or single parking space owner. Numerical experiments were conducted to illustrate the proposed concepts and assess the potential impact of crowdsourced parking spaces on the parking market. The results show that social welfare increases by more than 5% when crowdsourced parking locations account for 7% of the parking market. Results from additional numerical experiments show that ignoring stochasticity results in revenue loss for all parking owners. The developed techniques aim to facilitate existing and new parking facility owners to participate in crowdsourced event-parking markets.

Market Design Considerations for Scarcity Pricing: A Stochastic Equilibrium Framework

Scarcity pricing is a mechanism for improving the valuation of reserve capacity in real-time electricity markets. The goal of scarcity pricing is to mitigate the missing money problem and enhance investment in flexible resources. The implementation of scarcity pricing is underway in a number of U.S. markets, including Texas and PJM. The implementation is also currently under consideration in Belgium. As the mechanism was originally conceived in the context of a U.S.-style two-settlement system, its implementation in a European setting poses a number of interesting market design dilemmas which can affect the back-propagation of scarcity prices to forward day-ahead markets for energy and reserve capacity. We propose a modeling framework for analyzing these market design choices based on stochastic equilibrium, and use this modeling framework in order to represent and analyze a wide range of market design proposals. We report results on a case study of the Belgian electricity market.

Integrating transit systems with ride-sourcing services: A study on the system users’ stochastic equilibrium problem

Ride-sourcing services are continuously gaining popularity among urban travelers. Due to their flexibility and responsiveness, transit agencies spot chances of collaborating with Transportation Network Companies (TNCs) in order to improve service quality, promote transit usage, and meanwhile lower their operational cost. In this paper, we studied an integrated transit system where ride-sourcing services complement the transit service as an efficient and economical access mode. The users of the integrated system consist of not only transit riders but also drivers for TNCs. Riders maximize their utilities by choosing whether and how to use the integrated system, and drivers maximize their payoffs by deciding how to serve riders. More importantly, in this two-sided system, their decisions are unfolded to each other and mutually influence one another such that an equilibrium is reached where no one can improve their outcome. We modeled the stochastic mode choice of riders and zone choice of drivers and captured the interactive attributes in a fixed-point problem. The existence of such an equilibrium was proved, and an iterative solution algorithm that utilizes network algorithms was proposed to find an equilibrium pattern of the stochastic demand-supply problem. A real-world case study using the Twin Cities’ data was carried out, and insights into the mechanism of the integrated transit system were developed. Various aspects of the system including user perception, demand estimation, pricing strategy, and subsidy strategy were investigated through the proposed model. These research outcomes contribute to a better comprehension of this promising mobility system and provide valuable knowledge on the planning and design of such systems.

Characterization of stochastic equilibrium controls by the Malliavin calculus

We derive a characterization of equilibrium controls in continuous-time, time-inconsistent control (TIC) problems using the Malliavin calculus. For this, the classical duality analysis of adjoint BSDEs is replaced with the Malliavin integration by parts. This results into a necessary and sufficient maximum principle which is applied to a linear-quadratic TIC problem, recovering previous results obtained by duality analysis in the mean-variance case, and extending them to the linear-quadratic setting. We also show that our results apply beyond the linear-quadratic case by treating the generalized Merton problem.

Stochastic Superiority Equilibrium in Game Theory

The definition of best response for a player in the Nash equilibrium is based on maximizing the expected utility given the strategy of the rest of the players in a game. In this work, we consider stochastic games, that is, games with random payoffs, in which a finite number of players engage only once or at most a limited number of times. In such games, players may choose to deviate from maximizing their expected utility. This is because maximizing expected utility strategy does not address the uncertainty in payoffs. We instead define a new notion of a stochastic superiority best response. This notion of best response results in a stochastic superiority equilibrium in which players choose to play the strategy that maximizes the probability of them being rewarded the most in a single round of the game rather than maximizing the expected received reward, subject to the actions of other players. We prove the stochastic superiority equilibrium to exist in all finite games, that is, games with a finite number of players and actions, and numerically compare its performance to Nash equilibrium in finite-time stochastic games. In certain cases, we show the payoff under the stochastic superiority equilibrium is 70% likely to be higher than the payoff under Nash equilibrium.

Coordination of power and natural gas markets via financial instruments

Current electricity and natural gas markets operate with deterministic description of uncertain supply, and in a temporally and sectorally decoupled way. This practice in energy systems is being challenged by the increasing integration of stochastic renewable energy sources. There is a growing need for exchanging operational flexibility among energy sectors, which requires to improve the sectoral coordination between electricity and natural gas markets. In addition, the dispatch of flexible units in both sectors needs to be made in a more uncertainty-aware manner, requiring to strengthen the temporal coordination between day-ahead and real-time energy markets. We explore the use of existing financial instruments in the form of virtual bidding (VB) as a market-based solution to enhance both sectoral and temporal coordination in energy markets. It is established in the literature that VB by purely financial players is able to enhance the temporal coordination between deterministic day-ahead and real-time markets. By developing various stochastic equilibrium and optimization models, we show that VB by physical players, i.e., gas-fired power plants, at the interface of power and natural gas systems is of great potential to improve not only the temporal coordination between deterministic day-ahead and real-time markets, but also the sectoral coordination between deterministic electricity and natural gas markets. We exploit a fully stochastic co-optimization model as an ideal benchmark, and numerically illustrate the benefits of VB for increasing the overall market efficiency in terms of reduced expected operational cost of the entire energy system. We eventually show that flexible resources in both electricity and natural gas markets are dispatched more efficiently in the day-ahead stage when VB exists.

Markov-game modeling of cyclist-pedestrian interactions in shared spaces: A multi-agent adversarial inverse reinforcement learning approach

Understanding and modeling road user dynamics and their microscopic interaction behaviour at shared space facilities are curial for several applications including safety and performance evaluations. Despite the multi-agent nature of road user interactions, the majority of previous studies modeled their interactions as a single-agent modeling framework, i.e., considering the other interaction agents as part of the passive environment. However, this assumption is unrealistic and could limit the model's accuracy and transferability in non-stationary road user environments. This study proposes a novel Multi-Agent Adversarial Inverse Reinforcement Learning approach (MA-AIRL) to model and simulate road user interactions at shared space facilities. Unlike the traditional game-theoretic framework that models multi-agent systems as a single time-step payoff, the proposed approach is based on Markov Games (MG) which models road users' sequential decisions concurrently. Moreover, the proposed model can handle bounded rationality agents, e.g., limited information access, through the implementation of the Logistic Stochastic Best Response Equilibrium (LSBRE) solution concept. The proposed algorithm recovers road users' multi-agent reward functions using adversarial deep neural network discriminators and estimates their optimal policies using Multi-agent Actor-Critic with Kronecker factors (MACK) deep reinforcement learning. Data from three shared space locations in Vancouver, BC and New York City, New York are used in this study. The model's performance is compared to a baseline single-agent Gaussian Process Inverse Reinforcement Learning (GPIRL). The results show that the multi-agent modeling framework led to a significantly more accurate prediction of road users’ behaviour and their evasive action mechanisms. Moreover, the recovered reward functions based on the single-agent modeling approach failed to capture the equilibrium solution concept similar to the multi-agent approach.

A performance measure analysis for traffic networks with random data and general monotone cost functions

We consider a congested traffic network where users behave according to the Wardrop equilibrium principle, but the data are uncertain and only known through their probability distributions. Within this framework, we propose a stochastic equilibrium model to analyse the network performance, which allows for nonlinear cost functions. The effectiveness of our approach is shown through numerical experiments on medium-size networks.

Gamma -Robust electricity market equilibrium models with transmission and generation investments

We consider uncertain robust electricity market equilibrium problems including transmission and generation investments. Electricity market equilibrium modeling has a long tradition but is, in most of the cases, applied in a deterministic setting in which all data of the model are known. Whereas there exist some literature on stochastic equilibrium problems, the field of robust equilibrium models is still in its infancy. We contribute to this new field of research by considering Gamma -robust electricity market equilibrium models on lossless DC networks with transmission and generation investments. We state the nominal market equilibrium problem as a mixed complementarity problem as well as its variational inequality and welfare optimization counterparts. For the latter, we then derive a Gamma -robust formulation and show that it is indeed the counterpart of a market equilibrium problem with robustified player problems. Finally, we present two case studies to gain insights into the general effects of robustification on electricity market models. In particular, our case studies reveal that the transmission system operator tends to act more risk-neutral in the robust setting, whereas generating firms clearly behave more risk-averse.

Stochastic bifurcation for two-time-scale dynamical system with α-stable Lévy noise

This work focuses on stochastic bifurcation for a slow–fast dynamical system driven by non-Gaussian α-stable Lévy noise. We prove the main result for the stochastic equilibrium states for the original system and the reduced system based on the random slow manifold. Then, it is verified that the slow reduced system bears the stochastic bifurcation phenomenon inherited from the original system. Furthermore, we investigate the number and stability type of stochastic equilibrium states for dynamical systems through numerical simulations, and it is illustrated that the slow reduced system captures the stochastic bifurcation of the original system.

A two stage cap-and-trade model with allowance re-trading and capacity investment: The case of the Chilean NDC targets

In this work, we study an alternative approach for capping and pricing carbon emissions in electric markets: the cap-and-trade paradigm with re-trade of allowances. We model the electric market (generators and allowances’ auctioneer) as a two stage stochastic capacity expansion equilibrium problem, where we allow future investment and re-trading of emission permits among generators. The model studies generation and future investments in the electric sector in two regimes of demand: deterministic and stochastic. The configuration enforces the reduction of carbon emissions by setting a carbon budget, which allows to assess the impact of green policies and pledges concerning an electric system. We use the proposed model to analyze the Chilean electric sector under a cap-and-trade paradigm as an alternative to the existing carbon tax. We show that the Chilean pledge regarding emissions reductions does not encourage a shift to greener technologies. Moreover, we characterize two strategies to comply with the renewable targets by mid-century. On the one hand, a stringent carbon budget that induces high price of carbon permits and phases out coal-based generators. On the other hand, a less stringent target which significantly encourages investment in renewable technologies, but with low remaining shares of coal-based electric generation towards 2050.

Output and welfare effect of green credit in China: Evidence from an estimated DSGE model

In China, green credit aims to guide the flow of credit funds by regulating the quantity and price of credit, so as to support the development of green industries and curb the emission behavior of polluting enterprises. Based on the behavior of microeconomic entities we construct a Dynamic Stochastic Genernal Equilibrium (DSGE) model to analyze the output and welfare effect of green credit in China. Specifically, the incentive green credit is taken as an example to carry out an empirical study, and we further quantitatively measure the output and welfare effect of green credit under different environmental regulations. We find that both price-based and quantity-based green credit have obvious output, environment, health and utility welfare effect, which is conducive to the green upgrading of industrial structure and can achieve a win-win situation of output and environment in China. However, there are some differences in transmission mechanism and utility welfare. In addition, in the short term, the environmental tax regulation inhibits the economic expansion effect of green credit, and in the long term, this inhibitory effect gradually disappears. The DSGE comprehensive evaluation framework constructed in this paper expands the application scope of DSGE model, and provides a quantitative theoretical model for the mechanism analysis of China’s green credit policy and similar policies. Furthermore this study provides theoretical and quantitative basis for the formulation of green credit policies and the improvement of macro-control policies in China.

Decomposition Algorithms for Some Deterministic and Two-Stage Stochastic Single-Leader Multi-Follower Games

We consider a certain class of hierarchical decision problems that can be viewed as single-leader multi-follower games, and be represented by a virtual market coordinator trying to set a price system for traded goods, according to some criterion that balances supply and demand. The objective function of the market coordinator involves the decisions of many agents, which are taken independently by solving convex optimization problems that depend on the price configuration and on realizations of future states of the economy. One traditional way of solving this problem is via a mixed complementarity formulation. However, this approach can become impractical when the numbers of agents and/or scenarios become large. This work concerns agent-wise and scenario-wise decomposition algorithms to solve the equilibrium problems in question, assuming that the solutions of the agents’ problems are unique, which is natural in many applications (when solutions are not unique, the approximating problems are still well-defined, but the convergence properties of the algorithm are not established). The algorithm is based on a previous work of the authors, where a suitable regularization of solution mappings of fully parameterized convex problems is developed. Here, we show one specific strategy to manage the regularization parameter, extend some theoretical results to the current setting, and prove that the smooth approximations of the market coordinator’s problem converge epigraphically to the original problem. Numerical experiments and some comparisons with the complementarity solver PATH are shown for the two-stage stochastic Walrasian equilibrium problem.

Inexact stochastic subgradient projection method for stochastic equilibrium problems with nonmonotone bifunctions: application to expected risk minimization in machine learning

This paper discusses a stochastic equilibrium problem for which the function is in the form of the expectation of nonmonotone bifunctions and the constraint set is closed and convex. This problem includes various applications such as stochastic variational inequalities, stochastic Nash equilibrium problems, and nonconvex stochastic optimization problems. For solving this stochastic equilibrium problem, we propose an inexact stochastic subgradient projection method. The proposed method sets a random realization of the bifunction and then updates its approximation by using both its stochastic subgradient and the projection onto the constraint set. The main contribution of this paper is to present a convergence analysis showing that, under certain assumptions, any accumulation point of the sequence generated by the proposed method using a constant step size almost surely belongs to the solution set of the stochastic equilibrium problem. A convergence rate analysis of the method is also provided to illustrate the method’s efficiency. Another contribution of this paper is to show that a machine learning algorithm based on the proposed method achieves the expected risk minimization for a class of least absolute selection and shrinkage operator (lasso) problems in statistical learning with sparsity. Numerical comparisons of the proposed machine learning algorithm with existing machine learning algorithms for the expected risk minimization using LIBSVM datasets demonstrate the effectiveness and superior classification accuracy of the proposed algorithm.